The peculiar (monic) polynomials, the zeros of which equal their coefficients

摘要

We evaluate the number of monic polynomials (of arbitrary degree $N$) thezeros of which equal their coefficients when these are allowed to takearbitrary complex values. In the following, we call polynomials with thisproperty {\em peculiar\/} polynomials. We further show that the problem ofdetermining the peculiar polynomials of degree $N$ simplifies when any of thecoefficients is either 0 or 1. We proceed to estimate the numbers of peculiarpolynomials of degree $N$ having one coefficient zero, or one coefficient equalto one, or neither.